waves – chapter 10.2 – 10.5
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Waves – Chapter 10.2 – 10.5• Transverse vs longitudinal• Frequency f, wavelength l, period T, wave speed – v = lf• Harmonic waves: y(x) = Asin(kx); with k = 2 /p l• Traveling waves: y(x, t) = Asin(kx +/– wt);
with w = 2pf• For waves on a string •Total energy is proportional to A2
v/
T
m L
Interference• This is a property of waves• Waves can pass through each other “like
ghosts” • Whenever two waves overlap in space, they add
together – or superpose – in a phenomenon called interference
• For harmonic waves of the same wavelength, when they are in phase they add together leading to constructive interference, and when they are out of phase by 180o – corresponding to l/2 – then they add together leading to destructive interference
Superposition of two equal amplitude harmonic waves
y A kx t and y A kx t1 2 sin( ) sin( )
y y y A kx t kx t 1 2 sin( ) sin( ) .
sin + sin = 2 sin ½(+) cos ½(-)
y A kx t 2 12
12cos sin
Beats
-3
-2
-1
0
1
2
3
0 20 40 60 80 100 120
time
amp
litu
de
Waves at a Boundary
time
position
Standing Waves• Superposition of equal amplitude waves
traveling in opposite directions
• Using the same trig identity
• No longer a traveling wave – but a standing wave
y A kx t and y A kx t1 2 sin( ) sin( )
1 2 2 sin cosy y y A kx t
string distance
amp
litu
de
string distance
amp
litu
de
Standing waves on a string• Only certain frequencies will allow standing
waves – we require the wave to reflect and return to the starting point in phase with another oscillation.
• Lowest such resonant frequency (also known as the first harmonic) is the fundamental with l/2 = L so
f1 = v/ = l v/2L• Next is the second harmonic with f2 = v/L since l
= L - introduces one node• In general l = 2L/n and fn = nv/2L = nf1 - with
n-1 nodes
Problem
• Ex. 10.3 A steel guitar string with a 10 gram mass and a total length of 1m has a length of 70 cm between the two fixed points. If the string is tuned to play an E at 330 Hz, find the tension in the string.
Standing Wave Resonance in the sand
Resonance
• Increase in energy input due to a matching of frequencies –
• other examples include NMR, ESR, resonance in springs, in pendula, in sound (we’ll see this one next)
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